Filename: 327-pow-over-intro.txt
Title: A First Take at PoW Over Introduction Circuits
Author: George Kadianakis, Mike Perry, David Goulet, tevador
Created: 2 April 2020
Status: Draft

0. Abstract

  This proposal aims to thwart introduction flooding DoS attacks by introducing
  a dynamic Proof-Of-Work protocol that occurs over introduction circuits.

1. Motivation

  So far our attempts at limiting the impact of introduction flooding DoS
  attacks on onion services has been focused on horizontal scaling with
  Onionbalance, optimizing the CPU usage of Tor and applying congestion control
  using rate limiting. While these measures move the goalpost forward, a core
  problem with onion service DoS is that building rendezvous circuits is a
  costly procedure both for the service and for the network. For more
  information on the limitations of rate-limiting when defending against DDoS,
  see [REF_TLS_1].

  If we ever hope to have truly reachable global onion services, we need to
  make it harder for attackers to overload the service with introduction
  requests. This proposal achieves this by allowing onion services to specify
  an optional dynamic proof-of-work scheme that its clients need to participate
  in if they want to get served.

  With the right parameters, this proof-of-work scheme acts as a gatekeeper to
  block amplification attacks by attackers while letting legitimate clients
  through.

1.1. Related work

  For a similar concept, see the three internet drafts that have been proposed
  for defending against TLS-based DDoS attacks using client puzzles [REF_TLS].

1.2. Threat model [THREAT_MODEL]

1.2.1. Attacker profiles [ATTACKER_MODEL]

  This proposal is written to thwart specific attackers. A simple PoW proposal
  cannot defend against all and every DoS attack on the Internet, but there are
  adverary models we can defend against.

  Let's start with some adversary profiles:

  "The script-kiddie"

    The script-kiddie has a single computer and pushes it to its
    limits. Perhaps it also has a VPS and a pwned server. We are talking about
    an attacker with total access to 10 Ghz of CPU and 10 GBs of RAM. We
    consider the total cost for this attacker to be zero $.

  "The small botnet"

    The small botnet is a bunch of computers lined up to do an introduction
    flooding attack. Assuming 500 medium-range computers, we are talking about
    an attacker with total access to 10 Thz of CPU and 10 TB of RAM. We consider
    the upfront cost for this attacker to be about $400.

  "The large botnet"

    The large botnet is a serious operation with many thousands of computers
    organized to do this attack. Assuming 100k medium-range computers, we are
    talking about an attacker with total access to 200 Thz of CPU and 200 TB of
    RAM. The upfront cost for this attacker is about $36k.

  We hope that this proposal can help us defend against the script-kiddie
  attacker and small botnets. To defend against a large botnet we would need
  more tools in our disposal (see [FUTURE_DESIGNS]).

1.2.2. User profiles [USER_MODEL]

  We have attackers and we have users. Here are a few user profiles:

  "The standard web user"

    This is a standard laptop/desktop user who is trying to browse the
    web. They don't know how these defences work and they don't care to
    configure or tweak them. They are gonna use the default values and if the
    site doesn't load, they are gonna close their browser and be sad at Tor.
    They run a 2Ghz computer with 4GB of RAM.

  "The motivated user"

    This is a user that really wants to reach their destination. They don't
    care about the journey; they just want to get there. They know what's going
    on; they are willing to tweak the default values and make their computer do
    expensive multi-minute PoW computations to get where they want to be.

  "The mobile user"

    This is a motivated user on a mobile phone. Even tho they want to read the
    news article, they don't have much leeway on stressing their machine to do
    more computation.

  We hope that this proposal will allow the motivated user to always connect
  where they want to connect to, and also give more chances to the other user
  groups to reach the destination.

1.2.3. The DoS Catch-22 [CATCH22]

  This proposal is not perfect and it does not cover all the use cases. Still,
  we think that by covering some use cases and giving reachability to the
  people who really need it, we will severely demotivate the attackers from
  continuing the DoS attacks and hence stop the DoS threat all
  together. Furthermore, by increasing the cost to launch a DoS attack, a big
  class of DoS attackers will disappear from the map, since the expected ROI
  will decrease.

2. System Overview

2.1. Tor protocol overview

                                          +----------------------------------+
                                          |          Onion Service           |
   +-------+ INTRO1  +-----------+ INTRO2 +--------+                         |
   |Client |-------->|Intro Point|------->|  PoW   |-----------+             |
   +-------+         +-----------+        |Verifier|           |             |
                                          +--------+           |             |
                                          |                    |             |
                                          |                    |             |
                                          |         +----------v---------+   |
                                          |         |Intro Priority Queue|   |
                                          +---------+--------------------+---+
                                                           |  |  |
                                                Rendezvous |  |  |
                                                  circuits |  |  |
                                                           v  v  v



  The proof-of-work scheme specified in this proposal takes place during the
  introduction phase of the onion service protocol.

  The system described in this proposal is not meant to be on all the time, and
  should only be enabled by services when under duress. The percentage of
  clients receiving puzzles can also be configured based on the load of the
  service.

  In summary, the following steps are taken for the protocol to complete:

  1) Service encodes PoW parameters in descriptor [DESC_POW]
  2) Client fetches descriptor and computes PoW [CLIENT_POW]
  3) Client completes PoW and sends results in INTRO1 cell [INTRO1_POW]
  4) Service verifies PoW and queues introduction based on PoW effort [SERVICE_VERIFY]

2.2. Proof-of-work overview

2.2.1. Primitives

  For our proof-of-work function we will use the 'equix' scheme by tevador
  [REF_EQUIX].  Equix is an assymetric PoW function based on Equihash<60,3>. It
  features lightning fast verification speed, and also aims to minimize the
  assymetry between CPU and GPU. Furthermore, it's designed for this particular
  use-case and hence cryptocurrency miners are not incentivized to make
  optimized ASICs for it.

  The Equix scheme provides two functions that will be used in this proposal:
      - equix_solve(seed, nonce, effort) which solves a puzzle instance.
      - equix_verify(seed, nonce, effort, solution) which verifies a puzzle solution.

  We tune equix in section [EQUIX_TUNING].

2.2.2. Dynamic PoW

  DoS is a dynamic problem where the attacker's capabilities constantly change,
  and hence we want our proof-of-work system to be dynamic and not stuck with a
  static difficulty setting. Hence, instead of forcing clients to go below a
  static target like in Bitcoin to be successful, we ask clients to "bid" using
  their PoW effort. Effectively, a client gets higher priority the higher
  effort they put into their proof-of-work. This is similar to how
  proof-of-stake works but instead of staking coins, you stake work.

  The benefit here is that legitimate clients who really care about getting
  access can spend a big amount of effort into their PoW computation, which
  should guarantee access to the service given reasonable adversary models. See
  [PARAM_TUNING] for more details about these guarantees and tradeoffs.

  As a way to improve reachability and UX, the service tries to estimate the
  effort needed for clients to get access at any given time and places it in
  the descriptor. See [EFFORT_ESTIMATION] for more details.

2.2.3. PoW effort

  For our dynamic PoW system to work, we will need to be able to compare PoW
  tokens with each other. To do so we define a function:
         unsigned effort(uint8_t *token)
  which takes as its argument a hash output token, interprets it as a
  bitstring, and returns the quotient of dividing a bitstring of 1s by it.

  So for example:
         effort(00000001100010101101) = 11111111111111111111 / 00000001100010101101
  or the same in decimal:
         effort(6317) = 1048575 / 6317 = 165.

  This definition of effort has the advantage of directly expressing the
  expected number of hashes that the client had to calculate to reach the
  effort. This is in contrast to the (cheaper) exponential effort definition of
  taking the number of leading zero bits.

3. Protocol specification

3.1. Service encodes PoW parameters in descriptor [DESC_POW]

  This whole protocol starts with the service encoding the PoW parameters in
  the 'encrypted' (inner) part of the v3 descriptor. As follows:

       "pow-params" SP type SP seed-b64 SP expiration-time NL

        [At most once]

        type: The type of PoW system used. We call the one specified here "v1"

        seed-b64: A random seed that should be used as the input to the PoW
                  hash function. Should be 32 random bytes encoded in base64
                  without trailing padding.

        suggested-effort: An unsigned integer specifying an effort value that
                  clients should aim for when contacting the service. See
                  [EFFORT_ESTIMATION] for more details here.

        expiration-time: A timestamp in "YYYY-MM-DD SP HH:MM:SS" format after
                         which the above seed expires and is no longer valid as
                         the input for PoW. It's needed so that the size of our
                         replay cache does not grow infinitely. It should be
                         set to RAND_TIME(now+7200, 900) seconds.

   The service should refresh its seed when expiration-time passes. The service
   SHOULD keep its previous seed in memory and accept PoWs using it to avoid
   race-conditions with clients that have an old seed. The service SHOULD avoid
   generating two consequent seeds that have a common 4 bytes prefix. See
   [INTRO1_POW] for more info.

   By RAND_TIME(ts, interval) we mean a time between ts-interval and ts, chosen
   uniformly at random.

3.2. Client fetches descriptor and computes PoW [CLIENT_POW]

  If a client receives a descriptor with "pow-params", it should assume that
  the service is expecting a PoW input as part of the introduction protocol.

  The client parses the descriptor and extracts the PoW parameters. It makes
  sure that the <expiration-time> has not expired and if it has, it needs to
  fetch a new descriptor.

  The client should then extract the <suggested-effort> field to configure its
  PoW 'target' (see [REF_TARGET]). The client SHOULD NOT accept 'target' values
  that will cause an infinite PoW computation. {XXX: How to enforce this?}

  To complete the PoW the client follows the following logic:

      a) Client selects a target effort E.
      b) Client generates a random 16-byte nonce N.
      c) Client derives seed C by decoding 'seed-b64'.
      d) Client calculates S = equix_solve(C || N || E)
      e) Client calculates R = blake2b(C || N || E || S)
      f) Client checks if R * E <= UINT32_MAX.
        f1) If yes, success! The client can submit N, E, the first 4 bytes of C
    and S.
        f2) If no, fail! The client interprets N as a 16-byte little-endian
    integer, increments it by 1 and goes back to step d).

  At the end of the above procedure, the client should have S as the solution
  of the Equix puzzle with N as the nonce, C as the seed. How quickly this
  happens depends solely on the target effort E parameter.

3.3. Client sends PoW in INTRO1 cell [INTRO1_POW]

  Now that the client has an answer to the puzzle it's time to encode it into
  an INTRODUCE1 cell. To do so the client adds an extension to the encrypted
  portion of the INTRODUCE1 cell by using the EXTENSIONS field (see
  [PROCESS_INTRO2] section in rend-spec-v3.txt). The encrypted portion of the
  INTRODUCE1 cell only gets read by the onion service and is ignored by the
  introduction point.

  We propose a new EXT_FIELD_TYPE value:

     [01] -- PROOF_OF_WORK

   The EXT_FIELD content format is:

        POW_VERSION    [1 byte]
        POW_NONCE      [16 bytes]
        POW_EFFORT     [4 bytes]
        POW_SEED       [4 bytes]
        POW_SOLUTION   [16 bytes]

   where:

    POW_VERSION is 1 for the protocol specified in this proposal
    POW_NONCE is the nonce 'N' from the section above
    POW_SEED is the first 4 bytes of the seed used

   This will increase the INTRODUCE1 payload size by 43 bytes since the
   extension type and length is 2 extra bytes, the N_EXTENSIONS field is always
   present and currently set to 0 and the EXT_FIELD is 41 bytes. According to
   ticket #33650, INTRODUCE1 cells currently have more than 200 bytes
   available.

3.4. Service verifies PoW and handles the introduction  [SERVICE_VERIFY]

   When a service receives an INTRODUCE1 with the PROOF_OF_WORK extension, it
   should check its configuration on whether proof-of-work is required to
   complete the introduction. If it's not required, the extension SHOULD BE
   ignored. If it is required, the service follows the procedure detailed in
   this section.

   If the service requires the PROOF_OF_WORK extension but received an
   INTRODUCE1 cell without any embedded proof-of-work, the service SHOULD
   consider this cell as a zero-effort introduction for the purposes of the
   priority queue (see section [INTRO_QUEUE]).

3.4.1. PoW verification [POW_VERIFY]

   To verify the client's proof-of-work the service MUST do the following steps:

      a) Find a valid seed C that starts with POW_SEED. Fail if no such seed
         exists.
      b) Fail if E = POW_EFFORT is lower than the minimum effort.
      c) Fail if N = POW_NONCE is present in the replay cache (see [REPLAY_PROTECTION[)
      d) Calculate R = blake2b(C || N || E || S)
      e) Fail if R * E > UINT32_MAX
      f) Fail if equix_verify(C || N || E, S) != EQUIX_OK
      g) Put the request in the queue with a priority of E

   If any of these steps fail the service MUST ignore this introduction request
   and abort the protocol.

   In this proposal we call the above steps the "top half" of introduction
   handling. If all the steps of the "top half" have passed, then the circuit
   is added to the introduction queue as detailed in section [INTRO_QUEUE].

3.4.1.1. Replay protection [REPLAY_PROTECTION]

  The service MUST NOT accept introduction requests with the same (seed, nonce)
  tuple. For this reason a replay protection mechanism must be employed.

  The simplest way is to use a simple hash table to check whether a (seed,
  nonce) tuple has been used before for the actiev duration of a
  seed. Depending on how long a seed stays active this might be a viable
  solution with reasonable memory/time overhead.

  If there is a worry that we might get too many introductions during the
  lifetime of a seed, we can use a Bloom filter as our replay cache
  mechanism. The probabilistic nature of Bloom filters means that sometimes we
  will flag some connections as replays even if they are not; with this false
  positive probability increasing as the number of entries increase. However,
  with the right parameter tuning this probability should be negligible and
  well handled by clients. {TODO: Figure bloom filter}

3.4.2. The Introduction Queue  [INTRO_QUEUE]

3.4.2.1. Adding introductions to the introduction queue [ADD_QUEUE]

  When PoW is enabled and a verified introduction comes through, the service
  instead of jumping straight into rendezvous, queues it and prioritizes it
  based on how much effort was devoted by the client to PoW. This means that
  introduction requests with high effort should be prioritized over those with
  low effort.

  To do so, the service maintains an "introduction priority queue" data
  structure. Each element in that priority queue is an introduction request,
  and its priority is the effort put into its PoW:

  When a verified introduction comes through, the service uses the effort()
  function with the solution S as its input, and uses the output to place requests
  into the right position of the priority_queue: The bigger the effort, the
  more priority it gets in the queue. If two elements have the same effort, the
  older one has priority over the newer one.

3.4.2.2. Handling introductions from the introduction queue [HANDLE_QUEUE]

  The service should handle introductions by pulling from the introduction
  queue. We call this part of introduction handling the "bottom half" because
  most of the computation happens in this stage. For a description of how we
  expect such a system to work in Tor, see [TOR_SCHEDULER] section.

3.4.3. PoW effort estimation [EFFORT_ESTIMATION]

  The service starts with a default suggested-effort value of 5000 (see
  [EQUIX_DIFFICULTY] section for more info).

  Then during its operation the service continuously keeps track of the
  received PoW cell efforts to inform its clients of the effort they should put
  in their introduction to get service. The service informs the clients by
  using the <suggested-effort> field in the descriptor.

  Everytime the service handles or trims an introduction request from the
  priority queue in [HANDLE_QUEUE], the service adds the request's effort to a
  sorted list.

  Then every HS_UPDATE_PERIOD seconds (which is controlled through a consensus
  parameter and has a default value of 300 seconds) and while the DoS feature
  is enabled, the service updates its <suggested-effort> value as follows:

    1. Set TOTAL_EFFORT to the sum of the effort of all valid requests that
       have been received since the last HS descriptor update (this includes
       all handled requests, trimmed requests and requests still in the queue)

    2. Set SUGGESTED_EFFORT = TOTAL_EFFORT / (SVC_BOTTOM_CAPACITY * HS_UPDATE_PERIOD).
       The denominator above is the max number of requests that the service
       could have handled during that time.

    3. Set <suggested-effort> to max(MIN_EFFORT, SUGGESTED_EFFORT).

    During the above procedure we use the following default values:
      - MIN_EFFORT = 1000, as the result of a simulation experiment [REF_TEVADOR_SIM]
      - SVC_BOTTOM_CAPACITY = 100, which is the number of introduction requests
        that can be handled by the service per second. This was computed in
        [POW_DIFFICULTY_TOR] as 180, but we reduced it to 100 to account for
        slower computers and networks.

  The above algorithm is meant to balance the suggested effort based on the
  effort of all received requests. It attempts to dynamically adjust the
  suggested effort so that it increases when an attack is received, and tones
  down when the attack has stopped.

  It's worth noting that the suggested-effort is not a hard limit to the
  efforts that are accepted by the service, and it's only meant to serve as a
  guideline for clients to reduce the number of unsuccessful requests that get
  to the service. The service still adds requests with lower effort than
  suggested-effort to the priority queue in [ADD_QUEUE].

  Finally, the above algorithm will never reset back to zero suggested-effort,
  even if the attack is completely over. That's because in that case it would
  be impossible to determine the total computing power of connecting
  clients. Instead it will reset back to MIN_EFFORT, and the operator will have
  to manually shut down the anti-DoS mechanism.

  {XXX: SVC_BOTTOM_CAPACITY is static above and will not be accurate for all
  boxes. Ideally we should calculate SVC_BOTTOM_CAPACITY dynamically based on
  the resources of every onion service while the algorithm is running.}

3.4.3.1. Updating descriptor with new suggested effort

  Every HS_UPDATE_PERIOD seconds the service should upload a new descriptor
  with a new suggested-effort value.

  The service should avoid uploading descriptors too often to avoid overwheming
  the HSDirs. The service SHOULD NOT upload descriptors more often than
  HS_UPDATE_PERIOD. The service SHOULD NOT upload a new descriptor if the
  suggested-effort value changes by less than 15%.

  {XXX: Is this too often? Perhaps we can set different limits for when the
  difficulty goes up and different for when it goes down. It's more important
  to update the descriptor when the difficulty goes up.}

  {XXX: What attacks are possible here? Can the attacker intentionally hit this
  rate-limit and then influence the suggested effort so that clients do not
  learn about the new effort?}

4. Client behavior [CLIENT_BEHAVIOR]

  This proposal introduces a bunch of new ways where a legitimate client can
  fail to reach the onion service.

  Furthermore, there is currently no end-to-end way for the onion service to
  inform the client that the introduction failed. The INTRO_ACK cell is not
  end-to-end (it's from the introduction point to the client) and hence it does
  not allow the service to inform the client that the rendezvous is never gonna
  occur.

  For this reason we need to define some client behaviors to work around these
  issues.

4.1. Clients handling timeouts [CLIENT_TIMEOUT]

  Alice can fail to reach the onion service if her introduction request gets
  trimmed off the priority queue in [HANDLE_QUEUE], or if the service does not
  get through its priority queue in time and the connection times out.

  This section presents a heuristic method for the client getting service even
  in such scenarios.

  If the rendezvous request times out, the client SHOULD fetch a new descriptor
  for the service to make sure that it's using the right suggested-effort for
  the PoW and the right PoW seed. The client SHOULD NOT fetch service
  descriptors more often than every 'hs-pow-desc-fetch-rate-limit' seconds
  (which is controlled through a consensus parameter and has a default value of
  600 seconds).

  {XXX: Is this too rare? Too often?}

  When the client fetches a new descriptor, it should try connecting to the
  service with the new suggested-effort and PoW seed. If that doesn't work, it
  should double the effort and retry. The client should keep on
  doubling-and-retrying until it manages to get service, or its able to fetch a
  new descriptor again.

  {XXX: This means that the client will keep on spinning and
  doubling-and-retrying for a service under this situation. There will never be
  a "Client connection timed out" page for the user. Is this good? Is this bad?
  Should we stop doubling-and-retrying after some iterations? Or should we
  throw a custom error page to the user, and ask the user to stop spinning
  whenever they want?}

4.3. Other descriptor issues

  Another race condition here is if the service enables PoW, while a client has
  a cached descriptor. How will the client notice that PoW is needed? Does it
  need to fetch a new descriptor? Should there be another feedback mechanism?

5. Attacker strategies [ATTACK_META]

  Now that we defined our protocol we need to start tweaking the various
  knobs. But before we can do that, we first need to understand a few
  high-level attacker strategies to see what we are fighting against.

5.1.1. Overwhelm PoW verification (aka "Overwhelm top half") [ATTACK_TOP_HALF]

  A basic attack here is the adversary spamming with bogus INTRO cells so that
  the service does not have computing capacity to even verify the
  proof-of-work. This adversary tries to overwhelm the procedure in the
  [POW_VERIFY] section.

  That's why we need the PoW algorithm to have a cheap verification time so
  that this attack is not possible: we tune this PoW parameter in section
  [POW_TUNING_VERIFICATION].

5.1.2. Overwhelm rendezvous capacity (aka "Overwhelm bottom half") [ATTACK_BOTTOM_HALF]

  Given the way the introduction queue works (see [HANDLE_QUEUE]), a very
  effective strategy for the attacker is to totally overwhelm the queue
  processing by sending more high-effort introductions than the onion service
  can handle at any given tick. This adversary tries to overwhelm the procedure
  in the [HANDLE_QUEUE] section.

  To do so, the attacker would have to send at least 20 high-effort
  introduction cells every 100ms, where high-effort is a PoW which is above the
  estimated level of "the motivated user" (see [USER_MODEL]).

  An easier attack for the adversary, is the same strategy but with
  introduction cells that are all above the comfortable level of "the standard
  user" (see [USER_MODEL]). This would block out all standard users and only
  allow motivated users to pass.

5.1.3. Hybrid overwhelm strategy [ATTACK_HYBRID]

  If both the top- and bottom- halves are processed by the same thread, this
  opens up the possibility for a "hybrid" attack. Given the performance figures
  for the bottom half (0.31 ms/req.) and the top half (5.5 ms/req.), the
  attacker can optimally deny service by submitting 91 high-effort requests and
  1520 invalid requests per second. This will completely saturate the main loop
  because:

  0.31*(1520+91) ~ 0.5 sec.
  5.5*91         ~ 0.5 sec.

  This attack only has half the bandwidth requirement of [ATTACK_TOP_HALF] and
  half the compute requirement of [ATTACK_BOTTOM_HALF].

  Alternatively, the attacker can adjust the ratio between invalid and
  high-effort requests depending on their bandwidth and compute capabilities.

5.1.4. Gaming the effort estimation logic [ATTACK_EFFORT]

  Another way to beat this system is for the attacker to game the effort
  estimation logic (see [EFFORT_ESTIMATION]). Essentialy, there are two attacks
  that we are trying to avoid:

  - Attacker sets descriptor suggested-effort to a very high value effectively
    making it impossible for most clients to produce a PoW token in a
    reasonable timeframe.
  - Attacker sets descriptor suggested-effort to a very small value so that
    most clients aim for a small value while the attacker comfortably launches
    an [ATTACK_BOTTOM_HALF] using medium effort PoW (see [REF_TEVADOR_1])

5.1.4. Precomputed PoW attack

  The attacker may precompute many valid PoW nonces and submit them all at once
  before the current seed expires, overwhelming the service temporarily even
  using a single computer. The current scheme gives the attackers 4 hours to
  launch this attack since each seed lasts 2 hours and the service caches two
  seeds.

  An attacker with this attack might be aiming to DoS the service for a limited
  amount of time, or to cause an [ATTACK_EFFORT] attack.

6. Parameter tuning [POW_TUNING]

  There are various parameters in this PoW system that need to be tuned:

  We first start by tuning the time it takes to verify a PoW token. We do this
  first because it's fundamental to the performance of onion services and can
  turn into a DoS vector of its own. We will do this tuning in a way that's
  agnostic to the chosen PoW function.

  We will then move towards analyzing the default difficulty setting for our
  PoW system. That defines the expected time for clients to succeed in our
  system, and the expected time for attackers to overwhelm our system. Same as
  above we will do this in a way that's agnostic to the chosen PoW function.

  Finally, using those two pieces we will tune our PoW function and pick the
  right default difficulty setting. At the end of this section we will know the
  resources that an attacker needs to overwhelm the onion service, the
  resources that the service needs to verify introduction requests, and the
  resources that legitimate clients need to get to the onion service.

6.1. PoW verification [POW_TUNING_VERIFICATION]

  Verifying a PoW token is the first thing that a service does when it receives
  an INTRODUCE2 cell and it's detailed in section [POW_VERIFY]. This
  verification happens during the "top half" part of the process. Every
  milisecond spent verifying PoW adds overhead to the already existing "top
  half" part of handling an introduction cell. Hence we should be careful to
  add minimal overhead here so that we don't enable attacks like [ATTACK_TOP_HALF].

  During our performance measurements in [TOR_MEASUREMENTS] we learned that the
  "top half" takes about 0.26 msecs in average, without doing any sort of PoW
  verification. Using that value we compute the following table, that describes
  the number of cells we can queue per second (aka times we can perform the
  "top half" process) for different values of PoW verification time:

      +---------------------+-----------------------+--------------+
      |PoW Verification Time| Total "top half" time | Cells Queued |
      |                     |                       |  per second  |
      |---------------------|-----------------------|--------------|
      |    0     msec       |    0.26      msec     |    3846      |
      |    1     msec       |    1.26      msec     |    793       |
      |    2     msec       |    2.26      msec     |    442       |
      |    3     msec       |    3.26      msec     |    306       |
      |    4     msec       |    4.26      msec     |    234       |
      |    5     msec       |    5.26      msec     |    190       |
      |    6     msec       |    6.26      msec     |    159       |
      |    7     msec       |    7.26      msec     |    137       |
      |    8     msec       |    8.26      msec     |    121       |
      |    9     msec       |    9.26      msec     |    107       |
      |    10    msec       |    10.26     msec     |    97        |
      +---------------------+-----------------------+--------------+

  Here is how you can read the table above:

  - For a PoW function with a 1ms verification time, an attacker needs to send
    793 dummy introduction cells per second to succeed in a [ATTACK_TOP_HALF] attack.

  - For a PoW function with a 2ms verification time, an attacker needs to send
    442 dummy introduction cells per second to succeed in a [ATTACK_TOP_HALF] attack.

  - For a PoW function with a 10ms verification time, an attacker needs to send
    97 dummy introduction cells per second to succeed in a [ATTACK_TOP_HALF] attack.

  Whether an attacker can succeed at that depends on the attacker's resources,
  but also on the network's capacity.

  Our purpose here is to have the smallest PoW verification overhead possible
  that also allows us to achieve all our other goals.

  [Note that the table above is simply the result of a naive multiplication and
  does not take into account all the auxiliary overheads that happen every
  second like the time to invoke the mainloop, the bottom-half processes, or
  pretty much anything other than the "top-half" processing.

  During our measurements the time to handle INTRODUCE2 cells dominates any
  other action time: There might be events that require a long processing time,
  but these are pretty infrequent (like uploading a new HS descriptor) and
  hence over a long time they smooth out. Hence extrapolating the total cells
  queued per second based on a single "top half" time seems like good enough to
  get some initial intuition. That said, the values of "Cells queued per
  second" from the table above, are likely much smaller than displayed above
  because of all the auxiliary overheads.]

6.2. PoW difficulty analysis [POW_DIFFICULTY_ANALYSIS]

  The difficulty setting of our PoW basically dictates how difficult it should
  be to get a success in our PoW system. An attacker who can get many successes
  per second can pull a successfull [ATTACK_BOTTOM_HALF] attack against our
  system.

  In classic PoW systems, "success" is defined as getting a hash output below
  the "target". However, since our system is dynamic, we define "success" as an
  abstract high-effort computation.

  Our system is dynamic but we still need a default difficulty settings that
  will define the metagame and be used for bootstrapping the system. The client
  and attacker can still aim higher or lower but for UX purposes and for
  analysis purposes we do need to define a default difficulty.

6.2.1. Analysis based on adversary power

  In this section we will try to do an analysis of PoW difficulty without using
  any sort of Tor-related or PoW-related benchmark numbers.

  We created the table (see [REF_TABLE]) below which shows how much time a
  legitimate client with a single machine should expect to burn before they get
  a single success. The x-axis is how many successes we want the attacker to be
  able to do per second: the more successes we allow the adversary, the more
  they can overwhelm our introduction queue. The y-axis is how many machines
  the adversary has in her disposal, ranging from just 5 to 1000.

       ===============================================================
       |    Expected Time (in seconds) Per Success For One Machine   |
 ===========================================================================
 |                                                                          |
 |   Attacker Succeses        1       5       10      20      30      50    |
 |       per second                                                         |
 |                                                                          |
 |            5               5       1       0       0       0       0     |
 |            50              50      10      5       2       1       1     |
 |            100             100     20      10      5       3       2     |
 | Attacker   200             200     40      20      10      6       4     |
 |  Boxes     300             300     60      30      15      10      6     |
 |            400             400     80      40      20      13      8     |
 |            500             500     100     50      25      16      10    |
 |            1000            1000    200     100     50      33      20    |
 |                                                                          |
 ============================================================================

  Here is how you can read the table above:

  - If an adversary has a botnet with 1000 boxes, and we want to limit her to 1
    success per second, then a legitimate client with a single box should be
    expected to spend 1000 seconds getting a single success.

  - If an adversary has a botnet with 1000 boxes, and we want to limit her to 5
    successes per second, then a legitimate client with a single box should be
    expected to spend 200 seconds getting a single success.

  - If an adversary has a botnet with 500 boxes, and we want to limit her to 5
    successes per second, then a legitimate client with a single box should be
    expected to spend 100 seconds getting a single success.

  - If an adversary has access to 50 boxes, and we want to limit her to 5
    successes per second, then a legitimate client with a single box should be
    expected to spend 10 seconds getting a single success.

  - If an adversary has access to 5 boxes, and we want to limit her to 5
    successes per second, then a legitimate client with a single box should be
    expected to spend 1 seconds getting a single success.

  With the above table we can create some profiles for default values of our
  PoW difficulty. So for example, we can use the last case as the default
  parameter for Tor Browser, and then create three more profiles for more
  expensive cases, scaling up to the first case which could be hardest since
  the client is expected to spend 15 minutes for a single introduction.

6.2.2. Analysis based on Tor's performance [POW_DIFFICULTY_TOR]

  To go deeper here, we can use the performance measurements from
  [TOR_MEASUREMENTS] to get a more specific intuition on the default
  difficulty. In particular, we learned that completely handling an
  introduction cell takes 5.55 msecs in average. Using that value, we can
  compute the following table, that describes the number of introduction cells
  we can handle per second for different values of PoW verification:

      +---------------------+-----------------------+--------------+
      |PoW Verification Time| Total time to handle  | Cells handled|
      |                     |   introduction cell   |  per second  |
      |---------------------|-----------------------|--------------|
      |    0      msec      |    5.55        msec   |    180.18    |
      |    1      msec      |    6.55        msec   |    152.67    |
      |    2      msec      |    7.55        msec   |    132.45    |
      |    3      msec      |    8.55        msec   |    116.96    |
      |    4      msec      |    9.55        mesc   |    104.71    |
      |    5      msec      |    10.55       msec   |    94.79     |
      |    6      msec      |    11.55       msec   |    86.58     |
      |    7      msec      |    12.55       msec   |    79.68     |
      |    8      msec      |    13.55       msec   |    73.80     |
      |    9      msec      |    14.55       msec   |    68.73     |
      |    10     msec      |    15.55       msec   |    64.31     |
      +---------------------+-----------------------+--------------+

  Here is how you can read the table above:

  - For a PoW function with a 1ms verification time, an attacker needs to send
    152 high-effort introduction cells per second to succeed in a
    [ATTACK_BOTTOM_HALF] attack.

  - For a PoW function with a 10ms verification time, an attacker needs to send
    64 high-effort introduction cells per second to succeed in a
    [ATTACK_BOTTOM_HALF] attack.

  We can use this table to specify a default difficulty that won't allow our
  target adversary to succeed in an [ATTACK_BOTTOM_HALF] attack.

  Of course, when it comes to this table, the same disclaimer as in section
  [POW_TUNING_VERIFICATION] is valid. That is, the above table is just a
  theoretical extrapolation and we expect the real values to be much lower
  since they depend on auxiliary processing overheads, and on the network's
  capacity.

6.3. Tuning equix difficulty [EQUIX_DIFFICULTY]

  The above two sections were not depending on a particular PoW scheme. They
  gave us an intuition on the values we are aiming for in terms of verification
  speed and PoW difficulty. Now we need to make things concrete:

  As described in section [EFFORT_ESTIMATION] we start the service with a
  default suggested-effort value of 5000. Given the benchmarks of EquiX
  [REF_EQUIX] this should take about 2 to 3 seconds on a modern CPU.

  With this default difficulty setting and given the table in
  [POW_DIFFICULTY_ANALYSIS] this means that an attacker with 50 boxes will be
  able to get about 20 successful PoWs per second, and an attacker with 100
  boxes about 40 successful PoWs per second.

  Then using the table in [POW_DIFFICULTY_TOR] we can see that the number of
  attacker's successes is not enough to overwhelm the service through an
  [ATTACK_BOTTOM_HALF] attack. That is, an attacker would need to do about 152
  introductions per second to overwhelm the service, whereas they can only do
  40 with 100 boxes.

7. Discussion

7.1. UX

  This proposal has user facing UX consequences.

  Here is some UX improvements that don't need user-input:

  - Primarily, there should be a way for Tor Browser to display to users that
    additional time (and resources) will be needed to access a service that is
    under attack. Depending on the design of the system, it might even be
    possible to estimate how much time it will take.

  And here are a few UX approaches that will need user-input and have an
  increasing engineering difficulty. Ideally this proposal will not need
  user-input and the default behavior should work for almost all cases.

  a) Tor Browser needs a "range field" which the user can use to specify how
     much effort they want to spend in PoW if this ever occurs while they are
     browsing. The ranges could be from "Easy" to "Difficult", or we could try
     to estimate time using an average computer. This setting is in the Tor
     Browser settings and users need to find it.

  b) We start with a default effort setting, and then we use the new onion
     errors (see #19251) to estimate when an onion service connection has
     failed because of DoS, and only then we present the user a "range field"
     which they can set dynamically. Detecting when an onion service connection
     has failed because of DoS can be hard because of the lack of feedback (see
     [CLIENT_BEHAVIOR])

  c) We start with a default effort setting, and if things fail we
     automatically try to figure out an effort setting that will work for the
     user by doing some trial-and-error connections with different effort
     values. Until the connection succeeds we present a "Service is
     overwhelmed, please wait" message to the user.

7.2. Future work [FUTURE_WORK]

7.2.1. Incremental improvements to this proposal

  There are various improvements that can be done in this proposal, and while
  we are trying to keep this v1 version simple, we need to keep the design
  extensible so that we build more features into it. In particular:

  - End-to-end introduction ACKs

    This proposal suffers from various UX issues because there is no end-to-end
    mechanism for an onion service to inform the client about its introduction
    request. If we had end-to-end introduction ACKs many of the problems from
    [CLIENT_BEHAVIOR] would be aleviated. The problem here is that end-to-end
    ACKs require modifications on the introduction point code and a network
    update which is a lengthy process.

  - Multithreading scheduler

    Our scheduler is pretty limited by the fact that Tor has a single-threaded
    design. If we improve our multithreading support we could handle a much
    greater amount of introduction requests per second.

7.2.2. Future designs [FUTURE_DESIGNS]

  This is just the beginning in DoS defences for Tor and there are various
  futured designs and schemes that we can investigate. Here is a brief summary
  of these:

  "More advanced PoW schemes" -- We could use more advanced memory-hard PoW
         schemes like MTP-argon2 or Itsuku to make it even harder for
         adversaries to create successful PoWs. Unfortunately these schemes
         have much bigger proof sizes, and they won't fit in INTRODUCE1 cells.
         See #31223 for more details.

  "Third-party anonymous credentials" -- We can use anonymous credentials and a
         third-party token issuance server on the clearnet to issue tokens
         based on PoW or CAPTCHA and then use those tokens to get access to the
         service. See [REF_CREDS] for more details.

  "PoW + Anonymous Credentials" -- We can make a hybrid of the above ideas
         where we present a hard puzzle to the user when connecting to the
         onion service, and if they solve it we then give the user a bunch of
         anonymous tokens that can be used in the future. This can all happen
         between the client and the service without a need for a third party.

  All of the above approaches are much more complicated than this proposal, and
  hence we want to start easy before we get into more serious projects.

7.3. Environment

  We love the environment! We are concerned of how PoW schemes can waste energy
  by doing useless hash iterations. Here is a few reasons we still decided to
  pursue a PoW approach here:

  "We are not making things worse" -- DoS attacks are already happening and
      attackers are already burning energy to carry them out both on the
      attacker side, on the service side and on the network side. We think that
      asking legitimate clients to carry out PoW computations is not gonna
      affect the equation too much, since an attacker right now can very
      quickly cause the same damage that hundreds of legitimate clients do a
      whole day.

  "We hope to make things better" -- The hope is that proposals like this will
      make the DoS actors go away and hence the PoW system will not be used. As
      long as DoS is happening there will be a waste of energy, but if we
      manage to demotivate them with technical means, the network as a whole
      will less wasteful. Also see [CATCH22] for a similar argument.

8. Acknowledgements

  Thanks a lot to tevador for the various improvements to the proposal and for
  helping us understand and tweak the RandomX scheme.

  Thanks to Solar Designer for the help in understanding the current PoW
  landscape, the various approaches we could take, and teaching us a few neat
  tricks.

Appendix A.  Little-t tor introduction scheduler

  This section describes how we will implement this proposal in the "tor"
  software (little-t tor).

  The following should be read as if tor is an onion service and thus the end
  point of all inbound data.

A.1. The Main Loop [MAIN_LOOP]

  Tor uses libevent for its mainloop. For network I/O operations, a mainloop
  event is used to inform tor if it can read on a certain socket, or a
  connection object in tor.

  From there, this event will empty the connection input buffer (inbuf) by
  extracting and processing a cell at a time. The mainloop is single threaded
  and thus each cell is handled sequentially.

  Processing an INTRODUCE2 cell at the onion service means a series of
  operations (in order):

    1) Unpack cell from inbuf to local buffer.

    2) Decrypt cell (AES operations).

    3) Parse cell header and process it depending on its RELAY_COMMAND.

    4) INTRODUCE2 cell handling which means building a rendezvous circuit:
        i)  Path selection
        ii) Launch circuit to first hop.

    5) Return to mainloop event which essentially means back to step (1).

  Tor will read at most 32 cells out of the inbuf per mainloop round.

A.2. Requirements for PoW

  With this proposal, in order to prioritize cells by the amount of PoW work
  it has done, cells can _not_ be processed sequentially as described above.

  Thus, we need a way to queue a certain number of cells, prioritize them and
  then process some cell(s) from the top of the queue (that is, the cells that
  have done the most PoW effort).

  We thus require a new cell processing flow that is _not_ compatible with
  current tor design. The elements are:

    - Validate PoW and place cells in a priority queue of INTRODUCE2 cells (as
      described in section [INTRO_QUEUE]).

    - Defer "bottom half" INTRO2 cell processing for after cells have been
      queued into the priority queue.

A.3. Proposed scheduler [TOR_SCHEDULER]

  The intuitive way to address the A.2 requirements would be to do this
  simple and naive approach:

    1) Mainloop: Empty inbuf INTRODUCE2 cells into priority queue

    2) Process all cells in pqueue

    3) Goto (1)

  However, we are worried that handling all those cells before returning to the
  mainloop opens possibilities of attack by an adversary since the priority
  queue is not gonna be kept up to date while we process all those cells. This
  means that we might spend lots of time dealing with introductions that don't
  deserve it. See [BOTTOM_HALF_SCHEDULER] for more details.

  We thus propose to split the INTRODUCE2 handling into two different steps:
  "top half" and "bottom half" process, as also mentioned in [POW_VERIFY]
  section above.

A.3.1. Top half and bottom half scheduler

  The top half process is responsible for queuing introductions into the
  priority queue as follows:

    a) Unpack cell from inbuf to local buffer.

    b) Decrypt cell (AES operations).

    c) Parse INTRODUCE2 cell header and validate PoW.

    d) Return to mainloop event which essentially means step (1).

  The top-half basically does all operations of section [MAIN_LOOP] except from (4).

  An then, the bottom-half process is responsible for handling introductions
  and doing rendezvous. To achieve this we introduce a new mainloop event to
  process the priority queue _after_ the top-half event has completed. This new
  event would do these operations sequentially:

    a) Pop INTRODUCE2 cell from priority queue.

    b) Parse and process INTRODUCE2 cell.

    c) End event and yield back to mainloop.

A.3.2. Scheduling the bottom half process [BOTTOM_HALF_SCHEDULER]

  The question now becomes: when should the "bottom half" event get triggered
  from the mainloop?

  We propose that this event is scheduled in when the network I/O event
  queues at least 1 cell into the priority queue. Then, as long as it has a
  cell in the queue, it would re-schedule itself for immediate execution
  meaning at the next mainloop round, it would execute again.

  The idea is to try to empty the queue as fast as it can in order to provide a
  fast response time to an introduction request but always leave a chance for
  more cells to appear between cell processing by yielding back to the
  mainloop. With this we are aiming to always have the most up-to-date version
  of the priority queue when we are completing introductions: this way we are
  prioritizing clients that spent a lot of time and effort completing their PoW.

  If the size of the queue drops to 0, it stops scheduling itself in order to
  not create a busy loop. The network I/O event will re-schedule it in time.

  Notice that the proposed solution will make the service handle 1 single
  introduction request at every main loop event. However, when we do
  performance measurements we might learn that it's preferable to bump the
  number of cells in the future from 1 to N where N <= 32.

A.4 Performance measurements

  This section will detail the performance measurements we've done on tor.git
  for handling an INTRODUCE2 cell and then a discussion on how much more CPU
  time we can add (for PoW validation) before it badly degrades our
  performance.

A.4.1 Tor measurements [TOR_MEASUREMENTS]

  In this section we will derive measurement numbers for the "top half" and
  "bottom half" parts of handling an introduction cell.

  These measurements have been done on tor.git at commit
  80031db32abebaf4d0a91c01db258fcdbd54a471.

  We've measured several set of actions of the INTRODUCE2 cell handling process
  on Intel(R) Xeon(R) CPU E5-2650 v4. Our service was accessed by an array of
  clients that sent introduction requests for a period of 60 seconds.

  1. Full Mainloop Event

     We start by measuring the full time it takes for a mainloop event to
     process an inbuf containing INTRODUCE2 cells. The mainloop event processed
     2.42 cells per invocation on average during our measurements.

     Total measurements: 3279

       Min: 0.30 msec - 1st Q.: 5.47 msec - Median: 5.91 msec
       Mean: 13.43 msec - 3rd Q.: 16.20 msec - Max: 257.95 msec

  2. INTRODUCE2 cell processing (bottom-half)

     We also measured how much time the "bottom half" part of the process
     takes. That's the heavy part of processing an introduction request as seen
     in step (4) of the [MAIN_LOOP] section:

     Total measurements: 7931

       Min: 0.28 msec - 1st Q.: 5.06 msec - Median: 5.33 msec
       Mean: 5.29 msec - 3rd Q.: 5.57 msec - Max: 14.64 msec

  3. Connection data read (top half)

     Now that we have the above pieces, we can use them to measure just the
     "top half" part of the procedure. That's when bytes are taken from the
     connection inbound buffer and parsed into an INTRODUCE2 cell where basic
     validation is done.

     There is an average of 2.42 INTRODUCE2 cells per mainloop event and so we
     divide that by the full mainloop event mean time to get the time for one
     cell. From that we substract the "bottom half" mean time to get how much
     the "top half" takes:

        => 13.43 / (7931 / 3279) = 5.55
        => 5.55 - 5.29 = 0.26

        Mean: 0.26 msec

  To summarize, during our measurements the average number of INTRODUCE2 cells
  a mainloop event processed is ~2.42 cells (7931 cells for 3279 mainloop
  invocations).

  This means that, taking the mean of mainloop event times, it takes ~5.55msec
  (13.43/2.42) to completely process an INTRODUCE2 cell. Then if we look deeper
  we see that the "top half" of INTRODUCE2 cell processing takes 0.26 msec in
  average, whereas the "bottom half" takes around 5.33 msec.

  The heavyness of the "bottom half" is to be expected since that's where 95%
  of the total work takes place: in particular the rendezvous path selection
  and circuit launch.

A.2. References

    [REF_EQUIX]: https://github.com/tevador/equix
                 https://github.com/tevador/equix/blob/master/devlog.md
    [REF_TABLE]: The table is based on the script below plus some manual editing for readability:
                 https://gist.github.com/asn-d6/99a936b0467b0cef88a677baaf0bbd04
    [REF_BOTNET]: https://media.kasperskycontenthub.com/wp-content/uploads/sites/43/2009/07/01121538/ynam_botnets_0907_en.pdf
    [REF_CREDS]: https://lists.torproject.org/pipermail/tor-dev/2020-March/014198.html
    [REF_TARGET]: https://en.bitcoin.it/wiki/Target
    [REF_TLS]: https://www.ietf.org/archive/id/draft-nygren-tls-client-puzzles-02.txt
               https://tools.ietf.org/id/draft-nir-tls-puzzles-00.html
               https://tools.ietf.org/html/draft-ietf-ipsecme-ddos-protection-10
    [REF_TLS_1]: https://www.ietf.org/archive/id/draft-nygren-tls-client-puzzles-02.txt
    [REF_TEVADOR_1]: https://lists.torproject.org/pipermail/tor-dev/2020-May/014268.html
    [REF_TEVADOR_2]: https://lists.torproject.org/pipermail/tor-dev/2020-June/014358.html
    [REF_TEVADOR_SIM]: https://github.com/tevador/scratchpad/blob/master/tor-pow/effort_sim.md
